Why do universities fund Ph.D. students in the sciences?

A good question. Why would the school make this investment (which could be closer to $100K/year)?

1. As pointed out, if you are TAing/grading/teaching, you are providing services that the school charges undergraduates for.
2. The funding of a university is not entirely like that of a business. Some of the money is a direct investment in purely academic pursuits, especially that from grants. Educating people and doing basic research is part of what that money is allocated for.
3. Averaged over all the graduate students, the direct value that they provide to the university in terms of research which goes on to get grants/prestige/donations/patents is substantial.
4. There is a non-trivial chance that you will become a wealthy donor to the school.

Universities don't "fund" Ph.D. candidates. They pay them salaries - or what should be recognized as salaries - to do research. In more normal states (such as the Netherlands), nobody is trying to deny this fact, and PhD candidates are formally in a employer-employee legal relation with their university. In other states (such as the US, or rather individual states within the federation) there are on-again-off-again legal struggles regarding this question.

As an example, look for the US NLRB decisions 332-111 (October 31st, 2000, NYU) and later decision 342-42 (July 13 2004, Brown U) edit: and the recent and excellent 364-90 (August 23 2016, Columbia U).

By the way, even employment in teaching isn't always recognized, and occasionally (again, in the US specifically, but less so in recent years) universities try to pass off the teaching work as training/learning experience and not pay PhD candidates for it.

Now, of course it's not quite that simple: The relation of a PhD candidate and his/her university is not entirely the same as that of the line worker and the factory, or the typist and office etc. When you're in a PhD program you are still learning and acquiring skills; however, unlike an undergraduate student, you do this mostly by carrying out actual research work (and perhaps also teaching work). These two types of activity are what a university is supposed carry out, so you are significantly contributing to realizing the university's (ongoing) objectives. A PhD candidate is a trained professional in his general field already when s/he is inducted, and s/he gradually acquires expertise, hones skills, and trains in the research aspect of his/her discipline, as opposed to other applicative aspects of it.

I think another point worth considering is that two of the costs you list this as covering, tuition and fees, go back to the university. So while they might technically be losing money on paper by giving you these things, you could see it more as a form of creative accounting. They are not actually losing it, but rather not making it.

Depending on the university and how they operate, they might expect to regain more of the money you are given in your stipend via other means (housing, food, parking, etc.).

Once you account for the difference between on-paper spending on you and how much you are actually getting in real money, you can start explaining the rest of it as the fact you are in fact doing work for the university. Working as a TA, doing research, and other tasks involved in the Ph.D process.

It's worth considering that Ph.D students are in the same job market as other bachelors / masters holders, and that the university needs to compete with the non-academic market to an extent. Depending on the field, you might be foregoing a substantial salary by obtaining a Ph.D. Not everyone is necessarily going to be willing to do that, so offering a lot of freebies can encourage people who might go straight to business to remain in academia, at least briefly.

As such, it's probably more helpful to view this money as a combination of discounts and payment for TA/research work. Trying to build favor with you should you become a successful researcher no doubt plays a part as well.